Definition of a Curve

The notion of a curve is one of the most important notions in differential geometry. In antiquity this notion had no explicit mathematical definition. Euclid, for example, defines a curve as a "length without width". At this time many wonderful and interesting curves were discovered and studied; however, the idea of a general curve remained at a trivial, obvious level. Further technological progress required the development of natural science, especially the evolution of mechanics and mathematics. It was necessary to understand clearly the foundations of mathematics and, in particular, to construct an accurate definition of a curve. The coordinates method proposed by Descartes prepared the way for a general definition of curves; mathematicians contemporary to Descartes defined a plane curve given by an equation @(x,y) = 0 as a set of points such that their Cartesian coordinates satisfy this equation. Another idea arose in mechanics: a curve is imagined as the trace of some moving point, whose coordinates depend on the time t . Jordan proposed the following definition: a space curve is a set of points whose Cartesian coordinates X, y, z are continuous functions

of some parameter t varying inside a real axis segment (a, 6); in other words, a curve is defined as the image of a real axis segment under a continuous map into the space. This definition seemed to be natural, but in 1890 Peano constructed a continuous map of a segment (a, h) into the space such that the image of (a, h) under this map covered the whole square (we will consider Peano's example in one of the following chapters). In 1897 Klein remarked: "What is an arbitrary curve?. . . One may say that at present in mathematics there exists no more dark and more indefinite notion

than the mentioned one. The object, which we call a mass curve, is a strip, whose length is sufficiently great with respect to another strip's measures. But for a curve to be a subject of strong mathematical consideration, we must idealize a curve in the same way as a point is idealized. And here some difficulties arise. . . Let us turn to a proposition playing an essential role in Riemann's investigations into foundations of geometry: the space can be viewed as a three-dimensional continuous manifold.. . We start from a construction of some scale on a mass straight line; then we decompose the scaled line into smaller parts and continue this operation until it is realizable. After that we make the most important step from an experience to an axiom: we postulate that the correspondence between points and real numbers is valid not only empirically, hut also absolutely. . . ."